Pre-Algebra Examples

$\frac{1 + a z^{- 1} + c z^{- 2}}{(d - a b) z^{- 1} - c b z^{- 2}}$

Step 1

Simplify the numerator.

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Step 1.1

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{1 + a \frac{1}{z} + c z^{- 2}}{(d - a b) z^{- 1} - c b z^{- 2}}$

Step 1.2

Combine $a$ and $\frac{1}{z}$ .

$\frac{1 + \frac{a}{z} + c z^{- 2}}{(d - a b) z^{- 1} - c b z^{- 2}}$

Step 1.3

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{1 + \frac{a}{z} + c \frac{1}{z^{2}}}{(d - a b) z^{- 1} - c b z^{- 2}}$

Step 1.4

Combine $c$ and $\frac{1}{z^{2}}$ .

$\frac{1 + \frac{a}{z} + \frac{c}{z^{2}}}{(d - a b) z^{- 1} - c b z^{- 2}}$

Step 1.5

Write $1$ as a fraction with a common denominator.

$\frac{\frac{z}{z} + \frac{a}{z} + \frac{c}{z^{2}}}{(d - a b) z^{- 1} - c b z^{- 2}}$

Step 1.6

Combine the numerators over the common denominator.

$\frac{\frac{z + a}{z} + \frac{c}{z^{2}}}{(d - a b) z^{- 1} - c b z^{- 2}}$

Step 1.7

To write $\frac{z + a}{z}$ as a fraction with a common denominator, multiply by $\frac{z}{z}$ .

$\frac{\frac{z + a}{z} \cdot \frac{z}{z} + \frac{c}{z^{2}}}{(d - a b) z^{- 1} - c b z^{- 2}}$

Step 1.8

Write each expression with a common denominator of $z^{2}$ , by multiplying each by an appropriate factor of $1$ .

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Step 1.8.1

Multiply $\frac{z + a}{z}$ by $\frac{z}{z}$ .

$\frac{\frac{(z + a) z}{z \cdot z} + \frac{c}{z^{2}}}{(d - a b) z^{- 1} - c b z^{- 2}}$

Step 1.8.2

Multiply $z$ by $z$ .

$\frac{\frac{(z + a) z}{z^{2}} + \frac{c}{z^{2}}}{(d - a b) z^{- 1} - c b z^{- 2}}$

Step 1.9

Combine the numerators over the common denominator.

$\frac{\frac{(z + a) z + c}{z^{2}}}{(d - a b) z^{- 1} - c b z^{- 2}}$

Step 1.10

Simplify the numerator.

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Step 1.10.1

Apply the distributive property.

$\frac{\frac{z \cdot z + a z + c}{z^{2}}}{(d - a b) z^{- 1} - c b z^{- 2}}$

Step 1.10.2

Multiply $z$ by $z$ .

$\frac{\frac{z^{2} + a z + c}{z^{2}}}{(d - a b) z^{- 1} - c b z^{- 2}}$

Step 2

Simplify the denominator.

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Step 2.1

Factor $z^{- 2}$ out of $(d - a b) z^{- 1} - c b z^{- 2}$ .

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Step 2.1.1

Factor $z^{- 2}$ out of $(d - a b) z^{- 1}$ .

$\frac{\frac{z^{2} + a z + c}{z^{2}}}{z^{- 2} ((d - a b) z) - c b z^{- 2}}$

Step 2.1.2

Factor $z^{- 2}$ out of $- c b z^{- 2}$ .

$\frac{\frac{z^{2} + a z + c}{z^{2}}}{z^{- 2} ((d - a b) z) + z^{- 2} (- c b)}$

Step 2.1.3

Factor $z^{- 2}$ out of $z^{- 2} ((d - a b) z) + z^{- 2} (- c b)$ .

$\frac{\frac{z^{2} + a z + c}{z^{2}}}{z^{- 2} ((d - a b) z - c b)}$

Step 2.2

Apply the distributive property.

$\frac{\frac{z^{2} + a z + c}{z^{2}}}{z^{- 2} (d z - a b z - c b)}$

Step 2.3

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{\frac{z^{2} + a z + c}{z^{2}}}{\frac{1}{z^{2}} (d z - a b z - c b)}$

Step 3

Multiply the numerator by the reciprocal of the denominator.

$\frac{z^{2} + a z + c}{z^{2}} \cdot \frac{1}{\frac{1}{z^{2}} (d z - a b z - c b)}$

Step 4

Factor $\frac{1}{z^{2}}$ out of $\frac{1}{\frac{1}{z^{2}} (d z - a b z - c b)}$ .

$\frac{z^{2} + a z + c}{z^{2}} (z^{2} \frac{1}{d z - a b z - c b})$

Step 5

Cancel the common factor of $z^{2}$ .

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Step 5.1

Cancel the common factor.

$\frac{z^{2} + a z + c}{z^{2}} (z^{2} \frac{1}{d z - a b z - c b})$

Step 5.2

Rewrite the expression.

$(z^{2} + a z + c) \frac{1}{d z - a b z - c b}$

Step 6

Multiply $z^{2} + a z + c$ by $\frac{1}{d z - a b z - c b}$ .

$\frac{z^{2} + a z + c}{d z - a b z - c b}$

Step 7

Split the fraction $\frac{z^{2} + a z + c}{d z - a b z - c b}$ into two fractions.

$\frac{z^{2} + a z}{d z - a b z - c b} + \frac{c}{d z - a b z - c b}$

Step 8

Split the fraction $\frac{z^{2} + a z}{d z - a b z - c b}$ into two fractions.

$\frac{z^{2}}{d z - a b z - c b} + \frac{a z}{d z - a b z - c b} + \frac{c}{d z - a b z - c b}$